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Algebra, Functions, Graphs, and Vectors 5

A

B

A B

where denoteð the emptð setł also called the null set.

CoincidenŁ sets
Twm non-empty setð A and B are coincident if and only if, for
all elementð x:

x A ↔ x B

Cardinality
The cardinalitð of a set is defined as the number of elementð in
the set. The null set has cardinality zero. The set of people in
a city, starð in a galaxy, or atomð in the observable universe has
finite cardinality.
Most commonly used number setð have infinite cardinality.
Some number setð have cardinality that is denumerable; such
a set can be completely defined in termð of a sequence, even
thougà there might be infinitely many elementð in the set.
Some infinite number setð have non-denumerable cardinality;
such a set cannot be completely defined in termð of a sequence.

One-onł function
Let A and B be twm non-empty sets. Suppose that for every
member of A, a function f assignð some member of B. Let a 1
and a be memberð of A. Let b and b be memberð of B, such
2
1
2
that f assignð f(a ) b and f(a ) b . Then f is a one-one
1
2
2
1
function if and only if:
a a → b b
1 2 1 2
Onto function
A function f from set A tm setB is an ontà function if and only
if:

b B → f(a) b for some a A